Global strong solutions for the multi-dimensional compressible Oldroyd-B model

In this paper, we mainly focus on studying the multi-dimensional compressible Oldroyd-B model, which is derived by Barrett, Lu, and Süli (Comm. Math. Sci. 2017) through the micro-macro analysis of the compressible Navier-Stokes-Fokker-Planck system in the case of Hookean bead-spring chains. We provide a unified method to study the system with the background polymer number density $ \eta_{\infty}\ge 0 $, including the vanishing case and the nonvanishing case, and establish the global-in-time existence of the strong solution for the associated Cauchy problem when the initial data are small in the critical Besov spaces. Comparing Zhai-Li's result (J. Differential Equations 2021), we consider the system with the multi-dimensional case $ n\geq2 $ and the background polymer number density $ \eta_{\infty}\ge 0 $, not only $ n = 2,3 $ and $ \eta_{\infty} = 0 $. And we also extend the initial condition for the global existence of strong solutions by Zhai and Li (J. Differential Equations 2021) to a large class of the initial data, specifically, $ (\eta_0-\eta_{\infty})^l\in \dot{B}^{\frac{n}{2}-1}_{2,1}(\mathbb{R}^n) $ and $ (\mathbb{T}_0-\mathbb{T}_{\infty})^h\in\dot{B}^{\frac{n}{p}-1+s}_{p,1}(\mathbb{R}^n) $ with $ -1\le s\le 0 $ and $ s>1-\frac{2n}{p} $. Additionally, we also obtain the decay rates of the solution in Besov spaces.